where \(V:= (R_{\tilde{W}}+R^{\ast }_{\tilde{W}})/2.\)  In accordance with the Theorem 5 [1]}, we have
\(\lambda _{i}(V)\asymp \lambda _{i}(R_{H}),\) 
thus we have \(s_{i}(R_{\tilde{W}})\le C\lambda ^{1/2}_{i}(R_{H});\,s^{-1}_{i}(R_{\tilde{W}})\ge C\lambda ^{1/2}_{i}( H ),\)  the detailed proof of the latter fact see in the Theorem 7 [1]}. Using the monotonous property of the functions, we have
\(\frac{\ln ^{t } s^{-1}_{i}(R_{\tilde{W}})}{s^{-2}_{i}(R_{\tilde{W}})} \le C\cdot \frac{\ln ^{t} \lambda _{i}(H)}{ \lambda _{i}(H) } \le C\cdot \frac{\alpha _{i}}{i^{t }};\;\;\;\frac{\ln ^{t/2} s^{-1}_{i}(R_{\tilde{W}})}{s^{-1}_{i}(R_{\tilde{W}})}\le C\cdot \frac{\alpha _{i}}{i^{t/2}},\) 
